ghc-bignum-1.3: GHC BigNum library
Copyright (c) Sylvain Henry 2019(c) Herbert Valerio Riedel 2014 BSD3 sylvain@haskus.fr provisional non-portable (GHC Extensions) Safe-Inferred Haskell2010

GHC.Num.Integer

Description

The Integer type.

Synopsis

# Documentation

data Integer Source #

Arbitrary precision integers. In contrast with fixed-size integral types such as Int, the Integer type represents the entire infinite range of integers.

Integers are stored in a kind of sign-magnitude form, hence do not expect two's complement form when using bit operations.

If the value is small (fit into an Int), IS constructor is used. Otherwise Integer and IN constructors are used to store a BigNat representing respectively the positive or the negative value magnitude.

Invariant: Integer and IN are used iff value doesn't fit in IS

Constructors

 IS !Int# iff value in [minBound::Int, maxBound::Int] range IP !BigNat# iff value in ]maxBound::Int, +inf[ range IN !BigNat# iff value in ]-inf, minBound::Int[ range

#### Instances

Instances details
 Source # Instance detailsDefined in GHC.Num.Integer Methods Source # Instance detailsDefined in GHC.Num.Integer Methods

Check Integer invariants

Check Integer invariants

Integer Zero

Integer One

Create a positive Integer from a BigNat

Create a negative Integer from a BigNat

Create an Integer from a sign-bit and a BigNat

Convert an Integer into a sign-bit and a BigNat

Convert an Integer into a BigNat.

Return 0 for negative Integers.

Create an Integer from an Int#

Create an Integer from an Int

Truncates Integer to least-significant Int#

Truncates Integer to least-significant Int#

Convert a Word# into an Integer

Convert a Word into an Integer

Create a negative Integer with the given Word magnitude

Create an Integer from a sign and a Word magnitude

Truncate an Integer into a Word

Truncate an Integer into a Word

Convert a Natural into an Integer

Convert a list of Word into an Integer

Convert an Integer into a Natural

Return 0 for negative Integers.

Convert an Integer into a Natural

Return absolute value

Convert an Integer into a Natural

Throw an Underflow exception if input is negative.

Negative predicate

Negative predicate

Zero predicate

One predicate

Not-equal predicate.

Equal predicate.

Lower-or-equal predicate.

Lower predicate.

Greater predicate.

Greater-or-equal predicate.

Equal predicate.

Not-equal predicate.

Greater predicate.

Lower-or-equal predicate.

Lower predicate.

Greater-or-equal predicate.

Compare two Integer

Subtract one Integer from another.

Add two Integers

Multiply two Integers

Negate Integer.

One edge-case issue to take into account is that Int's range is not symmetric around 0. I.e. minBound+maxBound = -1

IP is used iff n > maxBound::Int IN is used iff n < minBound::Int

Compute absolute value of an Integer

Return -1, 0, and 1 depending on whether argument is negative, zero, or positive, respectively

Return -1#, 0#, and 1# depending on whether argument is negative, zero, or positive, respectively

Count number of set bits. For negative arguments returns the negated population count of the absolute value.

Positive Integer for which only n-th bit is set

Integer for which only n-th bit is set

Test if n-th bit is set.

Fake 2's complement for negative values (might be slow)

Test if n-th bit is set. For negative Integers it tests the n-th bit of the negated argument.

Fake 2's complement for negative values (might be slow)

Shift-right operation

Fake 2's complement for negative values (might be slow)

Shift-right operation

Fake 2's complement for negative values (might be slow)

Shift-left operation

Shift-left operation

Remember that bits are stored in sign-magnitude form, hence the behavior of negative Integers is different from negative Int's behavior.

Bitwise OR operation

Fake 2's complement for negative values (might be slow)

Bitwise XOR operation

Fake 2's complement for negative values (might be slow)

Bitwise AND operation

Fake 2's complement for negative values (might be slow)

Binary complement of the

Simultaneous integerQuot and integerRem.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Simultaneous integerQuot and integerRem.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Simultaneous integerDiv and integerMod.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Simultaneous integerDiv and integerMod.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Compute greatest common divisor.

Compute least common multiple.

Square a Integer

Base 2 logarithm (floor)

For numbers <= 0, return 0

Base 2 logarithm (floor)

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

integerIsPowerOf2# :: Integer -> (# (# #) | Word# #) Source #

Indicate if the value is a power of two and which one

Convert an Int64# into an Integer

Convert a Word64# into an Integer

Convert an Integer into an Int64#

Convert an Integer into a Word64#

Decode a Double# into (# Integer mantissa, Int# exponent #)

Encode (# Integer mantissa, Int# exponent #) into a Double#

Encode (Integer mantissa, Int exponent) into a Double

Encode (# Integer mantissa, Int# exponent #) into a Float#

TODO: Not sure if it's worth to write Float optimized versions here

Compute the number of digits of the Integer (without the sign) in the given base.

base must be > 1

integerToAddr# :: Integer -> Addr# -> Bool# -> State# s -> (# State# s, Word# #) Source #

Write an Integer (without sign) to addr in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: write most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Write an Integer (without sign) to addr in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: write most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

integerFromAddr# :: Word# -> Addr# -> Bool# -> State# s -> (# State# s, Integer #) Source #

Read an Integer (without sign) in base-256 representation from an Addr#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

Read an Integer (without sign) in base-256 representation from an Addr#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

integerToMutableByteArray# :: Integer -> MutableByteArray# s -> Word# -> Bool# -> State# s -> (# State# s, Word# #) Source #

Write an Integer (without sign) in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Write an Integer (without sign) in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

integerFromByteArray# :: Word# -> ByteArray# -> Word# -> Bool# -> State# s -> (# State# s, Integer #) Source #

Read an Integer (without sign) in base-256 representation from a ByteArray#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

Read an Integer (without sign) in base-256 representation from a ByteArray#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

Get the extended GCD of two integers.

integerGcde# a b returns (# g,x,y #) where * ax + by = g = |gcd a b|

Get the extended GCD of two integers.

integerGcde a b returns (g,x,y) where * ax + by = g = |gcd a b|

integerRecipMod# :: Integer -> Natural -> (# Natural | () #) Source #

Computes the modular inverse.

I.e. y = integerRecipMod# x m = x^(-1) mod m

with 0 < y < |m|

integerPowMod# :: Integer -> Integer -> Natural -> (# Natural | () #) Source #

Computes the modular exponentiation.

I.e. y = integer_powmod b e m = b^e mod m

with 0 <= y < abs m

If e is negative, we use integerRecipMod# to try to find a modular multiplicative inverse (which may not exist).